77 research outputs found
On Adaptivity Gaps of Influence Maximization Under the Independent Cascade Model with Full-Adoption Feedback
In this paper, we study the adaptivity gap of the influence maximization problem under the independent cascade model when full-adoption feedback is available. Our main results are to derive upper bounds on several families of well-studied influence graphs, including in-arborescences, out-arborescences and bipartite graphs. Especially, we prove that the adaptivity gap for the in-arborescences is between [e/(e-1), 2e/(e-1)], and for the out-arborescences the gap is between [e/(e-1), 2]. These are the first constant upper bounds in the full-adoption feedback model. Our analysis provides several novel ideas to tackle the correlated feedback appearing in adaptive stochastic optimization, which may be of independent interest
Complexity of Equilibria in First-Price Auctions under General Tie-Breaking Rules
We study the complexity of finding an approximate (pure) Bayesian Nash
equilibrium in a first-price auction with common priors when the tie-breaking
rule is part of the input. We show that the problem is PPAD-complete even when
the tie-breaking rule is trilateral (i.e., it specifies item allocations when
no more than three bidders are in tie, and adopts the uniform tie-breaking rule
otherwise). This is the first hardness result for equilibrium computation in
first-price auctions with common priors. On the positive side, we give a PTAS
for the problem under the uniform tie-breaking rule
Memory-Query Tradeoffs for Randomized Convex Optimization
We show that any randomized first-order algorithm which minimizes a
-dimensional, -Lipschitz convex function over the unit ball must either
use bits of memory or make
queries, for any constant and when the precision
is quasipolynomially small in . Our result implies that cutting plane
methods, which use bits of memory and queries,
are Pareto-optimal among randomized first-order algorithms, and quadratic
memory is required to achieve optimal query complexity for convex optimization
The complexity of non-stationary reinforcement learning
The problem of continual learning in the domain of reinforcement learning,
often called non-stationary reinforcement learning, has been identified as an
important challenge to the application of reinforcement learning. We prove a
worst-case complexity result, which we believe captures this challenge:
Modifying the probabilities or the reward of a single state-action pair in a
reinforcement learning problem requires an amount of time almost as large as
the number of states in order to keep the value function up to date, unless the
strong exponential time hypothesis (SETH) is false; SETH is a widely accepted
strengthening of the P NP conjecture. Recall that the number of states
in current applications of reinforcement learning is typically astronomical. In
contrast, we show that just a new state-action pair is
considerably easier to implement
Fast swap regret minimization and applications to approximate correlated equilibria
We give a simple and computationally efficient algorithm that, for any
constant , obtains -swap regret within only rounds; this is an exponential improvement compared to the
super-linear number of rounds required by the state-of-the-art algorithm, and
resolves the main open problem of [Blum and Mansour 2007]. Our algorithm has an
exponential dependence on , but we prove a new, matching lower
bound.
Our algorithm for swap regret implies faster convergence to
-Correlated Equilibrium (-CE) in several regimes: For
normal form two-player games with actions, it implies the first uncoupled
dynamics that converges to the set of -CE in polylogarithmic
rounds; a -bit communication protocol for -CE
in two-player games (resolving an open problem mentioned by
[Babichenko-Rubinstein'2017, Goos-Rubinstein'2018, Ganor-CS'2018]); and an
-query algorithm for -CE (resolving an open problem
of [Babichenko'2020] and obtaining the first separation between
-CE and -Nash equilibrium in the query complexity
model).
For extensive-form games, our algorithm implies a PTAS for
, a solution concept
often conjectured to be computationally intractable (e.g. [Stengel-Forges'08,
Fujii'23])
Primal-Dual Schemes for Online Matching in Bounded Degree Graphs
We explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem [Kalyanasundaram and Pruhs, 2000], the allocation problem [Buchbinder et al., 2007], and the AdWords problem [Mehta et al., 2007] in a beyond-worst-case setting. Specifically, we assume that G is a (k, d)-bounded degree graph, introduced by Naor and Wajc [Naor and Wajc, 2018]. Such graphs model natural properties on the degrees of advertisers and queries in the allocation and AdWords problems. While previous work only considers the scenario where k ? d, we consider the interesting intermediate regime of k ? d and prove a tight competitive ratio as a function of k,d (under the small-bid assumption) of ?(k,d) = 1 - (1-k/d)?(1-1/d)^{d - k} for the b-matching and allocation problems. We exploit primal-dual schemes [Buchbinder et al., 2009; Azar et al., 2017] to design and analyze the corresponding tight upper and lower bounds. Finally, we show a separation between the allocation and AdWords problems. We demonstrate that ?(k,d) competitiveness is impossible for the AdWords problem even in (k,d)-bounded degree graphs
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